A totally bounded uniformity on coarse metric spaces (Q2312468)

In this paper, a new version of boundary on coarse metric spaces is presented. For any metric space \(X\), the endpoints are defined as the images of coarse maps \(\mathbb{Z}_{+} \to X\) modulo finite Hausdorff distance, and a uniform structure (and a topological structure) on the set of endpoints, denoted by \(O(X)\), is defined by the coarse covers of \(X\). Every coarse map \(f: X\to Y\) induces a well-defined uniformly continuous map \(O(f): O(X) \to O(Y)\). The author proves that \(O\) defines a functor from the category \textsf{mCoarse} of metric spaces and coarse maps modulo closeness to the category \textsf{Top} of topological spaces and continuous maps and also to the category \textsf{Uniform} of uniform spaces and uniformly continuous maps. It is also shown that the uniform space \(O(X)\) is totally bounded and separating.